Tools for Signal Diagnosis

All symbol intervals are overlayed on top of one another and the time axis is shifted to bring ideal sampling instant in the middle. Eye diagram generated for 250 2-PAM symbols and Square-Root Raised Cosine pulse with excess bandwidth 0.5

We are now in a position to devise some tools that help us diagnose problems with the communication system under study. I like to call them the stethoscopes for a communications system due to the crucial functionality they provide regarding the health of the communication system being analyzed. We discuss two such tools, namely an eye diagram and a scatter plot below.

Eye Diagram


Imagine the samples of matched filter output in a stream of PAM symbols Figure taken at a much higher rate, say L=64 samples/symbol, instead of L=2 samples/symbol so that the underlying plot looks continuous as in Figure below. The waveform is divided into black boxes of width T_M seconds such that each signal portion within a box starts half a symbol duration T_M/2 before the ideal sampling time iT_M, where i is an integer. Similarly, each signal portion within a box ends half a symbol duration T_M/2 after the ideal sampling time iT_M as well.

Matched filter output for PAM as a continuous waveform

Assume that the such a stream of PAM symbols is printed on a paper and all black boxes are cut into separate pieces precisely at symbol boundaries. My daughter just did that when I gave her a printed PAM sequence, as shown below.

PAM symbol stream printed on a paper and cut into pieces at symbol boundaries

Now if they are placed on top of one another, we get a diagram drawn on the left in Figure below. This is eye diagram, which is a modulo-T_M plot of matched filter output against time. Compare the color patterns between both figures and observe that time base in left Figure below is shifted such that optimum sampling instant occurs in the middle of the plot. This shift in time base can sometimes cause confusion and hence it should be remembered that although the ideal sampling instants actually occur at the end of a symbol interval, eye diagram shows that instant in the middle.

All symbol intervals are overlayed on top of one another and the time axis is shifted to bring ideal sampling instant in the middle. Eye diagram generated for 250 2-PAM symbols and Square-Root Raised Cosine pulse with excess bandwidth 0.5

Also note that the eventual pattern strongly depends on the underlying pulse auto-correlation from which the pulse shape for transmitting data is derived. Now remember that we are only using an example of a 2-PAM modulation with 10 bits, and hence 10 symbols, only. When a large number of symbols are generated for such a plot and overlaid on each other, all possible trajectories of the pulse auto-correlation dictated by the symbol sequence come into play. A diagram for Square-Root Raised Cosine pulse shape (and hence Raised Cosine pulse auto-correlation) generated for 250 symbols and excess bandwidth \alpha=0.5 is shown on the right side of Figure above. Notice that this plot resembles a human eye, hence the name eye diagram.

Example


Interestingly, tracing a single transition in an eye diagram gives information about 3 symbols. As an example, look at the green trace around 5T_M on the left in Figure above:

[NOW] Observe that the trace goes through +A at the current sampling instant.
[Past] Also, it starts from a high voltage level. That is an indicator that in a clean system, its previous symbol would have been +A. Now compare it with matched filter output Figure for PAM and it is verified that its previous symbol, the purple trace, is indeed +A.
[Future] Finally, its level is falling below zero and towards -A which indicates that the next symbol should be -A. The fact that the black trace in matched filter output Figure for PAM is -A verifies this observation.

This will help us when we discuss symbol timing synchronization later.

The Purpose of Eye Diagram


Eye diagram is a diagnostic tool that helps in evaluation of the effects of channel noise and Inter-Symbol Interference (ISI) on the performance of a communication system. For this purpose, an eye diagram for a real transmission system can be generated through an oscilloscope. The horizontal time base of the oscilloscope is set equal to a symbol interval T_M and the matched filtered sequence is connected to the vertical axis. This superimposes the symbol intervals into a family of traces, all displayed within the same duration. The persistence of oscilloscope display makes it look like an eye.

To see how it helps, consider Figure below that consists of two symbol durations. Relevant information that can be extracted from an eye diagram is detailed below.

Eye diagram as a system diagnostic tool in a noiseless case

[Best Sampling Instant] For Nyquist pulses, no ISI occurs at the end of a symbol duration T_M — which falls in the center of the eye. This gives the maximum possible SNR on average because the sample value is farthest from the decision threshold at this point. A wrong decision will only happen if noise is sufficient to move the sample towards the other side of this threshold.
[Timing Error] A timing error occurs when the eye is not sampled at maximum average opening.
[Noise Margin] If a timing error occurs, the signal is sampled closer to the decision boundary. A relatively smaller amount of noise can cause a decision error. Or in other words, noise margin gets reduced. \footnote{For some trajectories, a timing error actually improves the noise margin, while for many others, it reduces that margin. When the average of all trajectories is taken into account, the noise margin decreases proportionally to the timing error.}
[Timing Jitter] Timing jitter is a measure of the average deviation of zero crossings.
[Eye Width] The wider the eye, the cleaner the channel. Channel distortion decreases the eye width and also decreases the eye opening at the best sampling instant. An open eye pattern corresponds to minimal signal distortion. Some channels can even cause the eye to close: there remains no sampling instant where a best symbol estimate can be obtained.

In fact, the eye in above Figure looks so symmetric because it is drawn for a noiseless case. Even for an AWGN channel with a low SNR, the eye can close but will remain symmetric in general as illustrated in Figure below. Depending on the SNR, there is distortion at the best sampling instant instead of zero ISI as in the previous case. Moreover, noise margin gets reduced, as a consequence of which decision errors can occur for relatively smaller noise power.

Eye diagram in an AWGN channel

Finally, channel distortions destroy this symmetry and result in strange looking eye diagrams, examples of which we will encounter in later chapters.

[Slope] Slope of the eye determines its sensitivity to timing errors. A large slope implies that even a little deviation in timing instant can cause the sample value — and subsequently noise margin — to reduce significantly. Therefore, a smaller slope lessens the effect of this dependence.

Eye diagrams can also be drawn for modulation schemes packing multiple bits in one symbol. For the case M=4, due to the presence of more than 2 symbols, the next symbol after every particular symbol can be +3A, +A, -A or -3A resulting into many different trajectories. Therefore, an eye diagram for modulation order M>2 displays multiple eyes, an example of which is illustrated in Figure below. Also notice that the larger the excess bandwidth \alpha, the lesser the interference among adjacent symbols in time and the eye is more open.

Eye diagrams generated for 400 4-PAM symbols and Square-Root Raised Cosine pulse for excess bandwidths 0.25 and 1

Scatter Plot


In general, a scatter plot is a graph that shows the relationship between two sets of data. For the purpose of digital communications, a scatter plot can be explained as follows.

Remember that we used a continuous version of PAM symbols Figure to understand eye diagrams. For scatter plot, we will use another PAM symbols Figure as a reference. After downsampling the matched filter output to symbol rate, the samples thus obtained are mapped back to the constellation as illustrated in above mentioned PAM symbols Figure. Before the symbol decisions are made, those samples form a cloud around the ideal constellation points as shown in Figure below. This cloud of symbol-spaced samples mapped on the original constellation diagram is called a scatter plot.

A scatter plot for binary PAM modulation

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